Here we are going through the ncert solutions for class 12 physics chapter 12– atoms. so before going through the ncert solutions make sure to go through the textbook that helps you to understand the solutions more easily

 ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​ ​​​​ Chapter 11 - Dual Nature Of Radiation And Matter


Question 11.1:

Find the

(a) maximum frequency, and

(b) minimum wavelength of X-rays produced by 30 kV electrons.


Potential of the electrons, V = 30 kV = 3 × 104 V

Hence, energy of the electrons, = 3 × 104 eV


e = Charge on an electron = 1.6 × 10−19 C

(a)Maximum frequency produced by the X-rays = ν

The energy of the electrons is given by the relation:


h = Planck’s constant = 6.626 × 10−34 Js


Hence, the maximum frequency of X-rays produced ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7454/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_m676e6075.gif

(b)The minimum wavelength produced by the X-rays is given as:


Hence, the minimum wavelength of X-rays produced is 0.0414 nm.


Question 11.2:

The work function of caesium metal is 2.14 eV. When light of frequency 6 ×1014 Hz is incident on the metal surface, photoemission of electrons occurs. What is the

(a) maximum kinetic energy of the emitted electrons,

(b) Stopping potential, and

(c) maximum speed of the emitted photoelectrons?


Work function of caesium metal, https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7457/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_6890acf.gif

Frequency of light, https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7457/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_655ba09c.gif

(a)The maximum kinetic energy is given by the photoelectric effect as:



h = Planck’s constant = 6.626 × 10−34 Js


Hence, the maximum kinetic energy of the emitted electrons is
0.345 eV.

(b)For stopping potentialhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7457/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_4e6ade8b.gif, we can write the equation for kinetic energy as:


Hence, the stopping potential of the material is 0.345 V.

(c)Maximum speed of the emitted photoelectrons = v

Hence, the relation for kinetic energy can be written as:



m = Mass of an electron = 9.1 × 10−31 kg


Hence, the maximum speed of the emitted photoelectrons is
332.3 km/s.


Question 11.3:

The photoelectric cut-off voltage in a certain experiment is 1.5 V. What is the maximum kinetic energy of photoelectrons emitted?


Photoelectric cut-off voltage, V0 = 1.5 V

The maximum kinetic energy of the emitted photoelectrons is given as:



e = Charge on an electron = 1.6 × 10−19 C


Therefore, the maximum kinetic energy of the photoelectrons emitted in the given experiment is 2.4 × 10−19 J.


Question 11.4:

Monochromatic light of wavelength 632.8 nm is produced by a helium-neon laser. The power emitted is 9.42 mW.

(a) Find the energy and momentum of each photon in the light beam,

(b) How many photons per second, on the average, arrive at a target irradiated by this beam? (Assume the beam to have uniform cross-section which is less than the target area), and

(c) How fast does a hydrogen atom have to travel in order to have the same momentum as that of the photon?


Wavelength of the monochromatic light, λ = 632.8 nm = 632.8 × 10−9 m

Power emitted by the laser, P = 9.42 mW = 9.42 × 10−3 W

Planck’s constant, h = 6.626 × 10−34 Js

Speed of light, c = 3 × 108 m/s

Mass of a hydrogen atom, m = 1.66 × 10−27 kg

(a)The energy of each photon is given as:


The momentum of each photon is given as:


(b)Number of photons arriving per second, at a target irradiated by the beam = n

Assume that the beam has a uniform cross-section that is less than the target area.

Hence, the equation for power can be written as:


(c)Momentum of the hydrogen atom is the same as the momentum of the photon, https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7461/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_m638ea585.gif

Momentum is given as:



v = Speed of the hydrogen atom



Question 11.5:

The energy flux of sunlight reaching the surface of the earth is 1.388 × 103 W/m2. How many photons (nearly) per square metre are incident on the Earth per second? Assume that the photons in the sunlight have an average wavelength of 550 nm.


Energy flux of sunlight reaching the surface of earth, Φ = 1.388 × 103 W/m2

Hence, power of sunlight per square metre, P = 1.388 × 103 W

Speed of light, c = 3 × 108 m/s

Planck’s constant, h = 6.626 × 10−34 Js

Average wavelength of photons present in sunlight, https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7462/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_m5d1e865c.gif


Number of photons per square metre incident on earth per second = n

Hence, the equation for power can be written as:


Therefore, every second, https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7462/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_m60aabf34.gif photons are incident per square metre on earth.


Question 11.6:

In an experiment on photoelectric effect, the slope of the cut-off voltage versus frequency of incident light is found to be 4.12 × 10−15 V s. Calculate the value of Planck’s constant.


The slope of the cut-off voltage (V) versus frequency (ν) of an incident light is given as:



e = Charge on an electron = 1.6 × 10−19 C

h = Planck’s constant


Therefore, the value of Planck’s constant is https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7464/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_42541e54.gif


Question 11.7:

A 100 W sodium lamp radiates energy uniformly in all directions. The lamp is located at the centre of a large sphere that absorbs all the sodium light which is incident on it. The wavelength of the sodium light is 589 nm. (a) What is the energy per photon associated with the sodium light? (b) At what rate are the photons delivered to the sphere?


Power of the sodium lamp, P = 100 W

Wavelength of the emitted sodium light, λ = 589 nm = 589 × 10−9 m

Planck’s constant, h = 6.626 × 10−34 Js

Speed of light, c = 3 × 108 m/s

(a)The energy per photon associated with the sodium light is given as:



(b)Number of photons delivered to the sphere = n

The equation for power can be written as:


Therefore, every second, https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7468/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_m5be7bdac.gifphotons are delivered to the sphere.

Question 11.8:

The threshold frequency for a certain metal is 3.3 × 1014 Hz. If light of frequency 8.2 × 1014 Hz is incident on the metal, predict the cutoff voltage for the photoelectric emission.


Threshold frequency of the metal, https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7473/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_41951640.gif

Frequency of light incident on the metal, https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7473/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_m793c714f.gif

Charge on an electron, e = 1.6 × 10−19 C

Planck’s constant, h = 6.626 × 10−34 Js

Cut-off voltage for the photoelectric emission from the metal = https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7473/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_4e6ade8b.gif

The equation for the cut-off energy is given as:


Therefore, the cut-off voltage for the photoelectric emission is https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7473/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_2a92effd.gif


Question 11.9:

The work function for a certain metal is 4.2 eV. Will this metal give photoelectric emission for incident radiation of wavelength 330 nm?



Work function of the metal, https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7478/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_383dfbf.gif

Charge on an electron, = 1.6 × 10−19 C

Planck’s constant, h = 6.626 × 10−34 Js

Wavelength of the incident radiation, λ = 330 nm = 330 × 10−9 m

Speed of light, c = 3 × 108 m/s

The energy of the incident photon is given as:


It can be observed that the energy of the incident radiation is less than the work function of the metal. Hence, no photoelectric emission will take place.


Question 11.10:

Light of frequency 7.21 × 1014 Hz is incident on a metal surface. Electrons with a maximum speed of 6.0 × 105 m/s are ejected from the surface. What is the threshold frequency for photoemission of electrons?


Frequency of the incident photon, https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7482/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_76b5353e.gif

Maximum speed of the electrons, v = 6.0 × 105 m/s

Planck’s constant, h = 6.626 × 10−34 Js

Mass of an electron, m = 9.1 × 10−31 kg

For threshold frequency ν0, the relation for kinetic energy is written as:


Therefore, the threshold frequency for the photoemission of electrons is https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7482/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_67b8ab15.gif


Question 11.11:

Light of wavelength 488 nm is produced by an argon laser which is used in the photoelectric effect. When light from this spectral line is incident on the emitter, the stopping (cut-off) potential of photoelectrons is 0.38 V. Find the work function of the material from which the emitter is made.


Wavelength of light produced by the argon laser, λ = 488 nm

= 488 × 10−9 m

Stopping potential of the photoelectrons, V0 = 0.38 V

1eV = 1.6 × 10−19 J

 V0 = https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7484/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_314db4b2.gif

Planck’s constant, h = 6.6 × 10−34 Js

Charge on an electron, e = 1.6 × 10−19 C

Speed of light, c = 3 × 10 m/s

From Einstein’s photoelectric effect, we have the relation involving the work function Φ0 of the material of the emitter as:


Therefore, the material with which the emitter is made has the work function of 2.16 eV.


Question 11.12:

Calculate the

(a) momentum, and

(b) de Broglie wavelength of the electrons accelerated through a potential difference of 56 V.


Potential difference, V = 56 V

Planck’s constant, h = 6.6 × 10−34 Js

Mass of an electron, m = 9.1 × 10−31 kg

Charge on an electron, e = 1.6 × 10−19 C

(a) At equilibrium, the kinetic energy of each electron is equal to the accelerating potential, i.e., we can write the relation for velocity (v) of each electron as:


The momentum of each accelerated electron is given as:

p = mv

= 9.1 × 10−31 × 4.44 × 106

= 4.04 × 10−24 kg m s−1

Therefore, the momentum of each electron is 4.04 × 10−24 kg m s−1.

(b) De Broglie wavelength of an electron accelerating through a potential V, is given by the relation:



Therefore, the de Broglie wavelength of each electron is 0.1639 nm.


Question 11.13:

What is the

(a) momentum,

(b) speed, and

(c) de Broglie wavelength of an electron with kinetic energy of 120 eV.


Kinetic energy of the electron, Ek = 120 eV

Planck’s constant, h = 6.6 × 10−34 Js

Mass of an electron, m = 9.1 × 10−31 kg

Charge on an electron, e = 1.6 × 10−19 C

(a) For the electron, we can write the relation for kinetic energy as:



v = Speed of the electron


Momentum of the electron, p = mv

= 9.1 × 10−31 × 6.496 × 106

= 5.91 × 10−24 kg m s−1

Therefore, the momentum of the electron is 5.91 × 10−24 kg m s−1.

(b) Speed of the electron, v = 6.496 × 106 m/s

(c) De Broglie wavelength of an electron having a momentum p, is given as:


Therefore, the de Broglie wavelength of the electron is 0.112 nm.


Question 11.14:

The wavelength of light from the spectral emission line of sodium is 589 nm. Find the kinetic energy at which

(a) an electron, and

(b) a neutron, would have the same de Broglie wavelength.


Wavelength of light of a sodium line, λ = 589 nm = 589 × 10−9 m

Mass of an electron, me= 9.1 × 10−31 kg

Mass of a neutron, mn= 1.66 × 10−27 kg

Planck’s constant, h = 6.6 × 10−34 Js

(a) For the kinetic energy K, of an electron accelerating with a velocity v, we have the relation:


We have the relation for de Broglie wavelength as:


Substituting equation (2) in equation (1), we get the relation:


Hence, the kinetic energy of the electron is 6.9 × 10−25 J or 4.31 μeV.

(b) Using equation (3), we can write the relation for the kinetic energy of the neutron as:




Hence, the kinetic energy of the neutron is 3.78 × 10−28 J or 2.36 neV.


Question 11.15:

What is the de Broglie wavelength of

(a) a bullet of mass 0.040 kg travelling at the speed of 1.0 km/s,

(b) a ball of mass 0.060 kg moving at a speed of 1.0 m/s, and

(c) a dust particle of mass 1.0 × 10−9 kg drifting with a speed of 2.2 m/s?


(a)Mass of the bullet, m = 0.040 kg

Speed of the bullet, v = 1.0 km/s = 1000 m/s

Planck’s constant, h = 6.6 × 10−34 Js

De Broglie wavelength of the bullet is given by the relation:



(b) Mass of the ball, m = 0.060 kg

Speed of the ball, v = 1.0 m/s

De Broglie wavelength of the ball is given by the relation:



(c)Mass of the dust particle, m = 1 × 10−9 kg

Speed of the dust particle, v = 2.2 m/s

De Broglie wavelength of the dust particle is given by the relation:




Question 11.16:

An electron and a photon each have a wavelength of 1.00 nm. Find

(a) their momenta,

(b) the energy of the photon, and

(c) the kinetic energy of electron.


Wavelength of an electron https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7495/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_m74ffdbc0.gif

= 1 × 10−9 m

Planck’s constant, h = 6.63 × 10−34 Js

(a) The momentum of an elementary particle is given by de Broglie relation:



It is clear that momentum depends only on the wavelength of the particle. Since the wavelengths of an electron and a photon are equal, both have an equal momentum.


(b) The energy of a photon is given by the relation:



Speed of light, c = 3 × 108 m/s


Therefore, the energy of the photon is 1.243 keV.

(c) The kinetic energy (K) of an electron having momentum p,is given by the relation:



m = Mass of the electron = 9.1 × 10−31 kg

= 6.63 × 10−25 kg m s−1


Hence, the kinetic energy of the electron is 1.51 eV.


Question 11.17:

(a) For what kinetic energy of a neutron will the associated de Broglie wavelength be 1.40 × 10−10 m?

(b) Also find the de Broglie wavelength of a neutron, in thermal equilibrium with matter, having an average kinetic energy of (3/2) kT at 300 K.


(a) De Broglie wavelength of the neutron, λ = 1.40 × 10−10 m

Mass of a neutron, mn = 1.66 × 10−27 kg

Planck’s constant, h = 6.6 × 10−34 Js

Kinetic energy (K) and velocity (v) are related as:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7496/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_m536f1bb5.gif … (1)

De Broglie wavelength (λ) and velocity (v) are related as:


Using equation (2) in equation (1), we get:


Hence, the kinetic energy of the neutron is 6.75 × 10−21 J or 4.219 × 10−2 eV.

(b) Temperature of the neutron, T = 300 K

Boltzmann constant, k = 1.38 × 10−23 kg m2 s−2 K−1

Average kinetic energy of the neutron:


The relation for the de Broglie wavelength is given as:


Therefore, the de Broglie wavelength of the neutron is 0.146 nm.


Question 11.18:

Show that the wavelength of electromagnetic radiation is equal to the de Broglie wavelength of its quantum (photon).


The momentum of a photon having energy () is given as:



λ = Wavelength of the electromagnetic radiation

c = Speed of light

h = Planck’s constant

De Broglie wavelength of the photon is given as:



m = Mass of the photon

v = Velocity of the photon

Hence, it can be inferred from equations (i) and (ii) that the wavelength of the electromagnetic radiation is equal to the de Broglie wavelength of the photon.


Question 11.19:

What is the de Broglie wavelength of a nitrogen molecule in air at 300 K? Assume that the molecule is moving with the root-mean square speed of molecules at this temperature. (Atomic mass of nitrogen = 14.0076 u)


Temperature of the nitrogen molecule, T = 300 K

Atomic mass of nitrogen = 14.0076 u

Hence, mass of the nitrogen molecule, m = 2 × 14.0076 = 28.0152 u

But 1 u = 1.66 × 10−27 kg

m = 28.0152 ×1.66 × 10−27 kg

Planck’s constant, h = 6.63 × 10−34 Js

Boltzmann constant, k = 1.38 × 10−23 J K−1

We have the expression that relates mean kinetic energy https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7500/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_30a34b1a.gif of the nitrogen molecule with the root mean square speed https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7500/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_m2d65f849.gifas:


Hence, the de Broglie wavelength of the nitrogen molecule is given as:


Therefore, the de Broglie wavelength of the nitrogen molecule is 0.028 nm.

Question 11.20:

(a) Estimate the speed with which electrons emitted from a heated emitter of an evacuated tube impinge on the collector maintained at a potential difference of 500 V with respect to the emitter. Ignore the small initial speeds of the electrons. The specific charge of the electron, i.e., its e/m is given to be 1.76 × 1011 C kg−1.

(b) Use the same formula you employ in (a) to obtain electron speed for an collector potential of 10 MV. Do you see what is wrong? In what way is the formula to be modified?


(a)Potential difference across the evacuated tube, V = 500 V

Specific charge of an electron, e/m = 1.76 × 1011 C kg−1

The speed of each emitted electron is given by the relation for kinetic energy as:


Therefore, the speed of each emitted electron is https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7502/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_m2382aa92.gif

(b)Potential of the anode, V = 10 MV = 10 × 106 V

The speed of each electron is given as:


This result is wrong because nothing can move faster than light. In the above formula, the expression (mv2/2) for energy can only be used in the non-relativistic limit, i.e., for v << c.

For very high speed problems, relativistic equations must be considered for solving them. In the relativistic limit, the total energy is given as:

E = mc2


= Relativistic mass


m0 = Mass of the particle at rest

Kinetic energy is given as:

K = mc2 − m0c2


Question 11.21:

(a) A monoenergetic electron beam with electron speed of 5.20 × 106 m s−1 is subject to a magnetic field of 1.30 × 10−4 T normal to the beam velocity. What is the radius of the circle traced by the beam, given e/m for electron equals 1.76 × 1011 C kg−1.

(b) Is the formula you employ in (a) valid for calculating radius of the path of a 20 MeV electron beam? If not, in what way is it modified?

[Note: Exercises 11.20(b) and 11.21(b) take you to relativistic mechanics which is beyond the scope of this book. They have been inserted here simply to emphasise the point that the formulas you use in part (a) of the exercises are not valid at very high speeds or energies. See answers at the end to know what ‘very high speed or energy’ means.]


(a)Speed of an electron, v = 5.20 × 106 m/s

Magnetic field experienced by the electron, B = 1.30 × 10−4 T

Specific charge of an electron, e/m = 1.76 × 1011 C kg−1


e = Charge on the electron = 1.6 × 10−19 C

m = Mass of the electron = 9.1 × 10−31 kg−1

The force exerted on the electron is given as:


θ = Angle between the magnetic field and the beam velocity

The magnetic field is normal to the direction of beam.


The beam traces a circular path of radius, r. It is the magnetic field, due to its bending nature, that provides the centripetal force https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7512/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_m71f47e3b.gif for the beam.

Hence, equation (1) reduces to:


Therefore, the radius of the circular path is 22.7 cm.

(b) Energy of the electron beam, = 20 MeVhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7512/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_6e837dce.gif

The energy of the electron is given as:


This result is incorrect because nothing can move faster than light. In the above formula, the expression (mv2/2) for energy can only be used in the non-relativistic limit, i.e., for v << c

When very high speeds are concerned, the relativistic domain comes into consideration.

In the relativistic domain, mass is given as:



https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7512/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_25e803f8.gif= Mass of the particle at rest

Hence, the radius of the circular path is given as:



Question 11.22:

An electron gun with its collector at a potential of 100 V fires out electrons in a spherical bulb containing hydrogen gas at low pressure (10−2 mm of Hg). A magnetic field of 2.83 × 10−4 T curves the path of the electrons in a circular orbit of radius 12.0 cm. (The path can be viewed because the gas ions in the path focus the beam by attracting electrons, and emitting light by electron capture; this method is known as the ‘fine beam tube’ method. Determine e/m from the data.


Potential of an anode, = 100 V

Magnetic field experienced by the electrons, B = 2.83 × 10−4 T

Radius of the circular orbit r = 12.0 cm = 12.0 × 10−2 m

Mass of each electron = m

Charge on each electron = e

Velocity of each electron = v

The energy of each electron is equal to its kinetic energy, i.e.,


It is the magnetic field, due to its bending nature, that provides the centripetal force https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7515/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_m71f47e3b.gif for the beam. Hence, we can write:

Centripetal force = Magnetic force


Putting the value of v in equation (1), we get:


Therefore, the specific charge ratio (e/m) ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7515/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_602347a8.gif


Question 11.23:

(a) An X-ray tube produces a continuous spectrum of radiation with its short wavelength end at 0.45 Å. What is the maximum energy of a photon in the radiation?

(b) From your answer to (a), guess what order of accelerating voltage (for electrons) is required in such a tube?


(a) Wavelength produced by an X-ray tube, https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7517/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_m58942c43.gif

Planck’s constant, h = 6.626 × 10−34 Js

Speed of light, = 3 × 108 m/s

The maximum energy of a photon is given as:


Therefore, the maximum energy of an X-ray photon is 27.6 keV.

(b) Accelerating voltage provides energy to the electrons for producing X-rays. To get an X-ray of 27.6 keV, the incident electrons must possess at least 27.6 keV of kinetic electric energy. Hence, an accelerating voltage of the order of 30 keV is required for producing X-rays.


Question 11.24:

In an accelerator experiment on high-energy collisions of electrons with positrons, a certain event is interpreted as annihilation of an electron-positron pair of total energy 10.2 BeV into two γ-rays of equal energy. What is the wavelength associated with each γ-ray? (1BeV = 109 eV)


Total energy of two γ-rays:

E = 10. 2 BeV

= 10.2 × 109 eV

= 10.2 × 109 × 1.6 × 10−10 J

Hence, the energy of each γ-ray:


Planck’s constant, https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7521/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_m2120543b.gif

Speed of light, https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7521/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_5da1d4a8.gif

Energy is related to wavelength as:


Therefore, the wavelength associated with each γ-ray ishttps://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7521/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_m22cf6b48.gif


Question 11.25:

Estimating the following two numbers should be interesting. The first number will tell you why radio engineers do not need to worry much about photons! The second number tells you why our eye can never ‘count photons’, even in barely detectable light.

(a) The number of photons emitted per second by a Medium wave transmitter of 10 kW power, emitting radiowaves of wavelength 500 m.

(b) The number of photons entering the pupil of our eye per second corresponding to the minimum intensity of white light that we humans can perceive (10−10 W m−2). Take the area of the pupil to be about 0.4 cm2, and the average frequency of white light to be about 6 × 1014 Hz.


(a) Power of the medium wave transmitter, P = 10 kW = 10W = 104 J/s

Hence, energy emitted by the transmitter per second, E = 104

Wavelength of the radio wave, λ = 500 m

The energy of the wave is given as:



h = Planck’s constant = 6.6 × 10−34 Js

c = Speed of light = 3 × 108 m/s


Let n be the number of photons emitted by the transmitter.

nE1 = E


The energy (E1) of a radio photon is very less, but the number of photons (n) emitted per second in a radio wave is very large.

The existence of a minimum quantum of energy can be ignored and the total energy of a radio wave can be treated as being continuous.

(b) Intensity of light perceived by the human eye, I = 10−10 W m−2

Area of a pupil, A = 0.4 cm= 0.4 × 10−4 m2

Frequency of white light, ν= 6 × 1014 Hz

The energy emitted by a photon is given as:

E = hν


h = Planck’s constant = 6.6 × 10−34 Js

E = 6.6 × 10−34 × 6 × 1014

= 3.96 × 10−19 J

Let n be the total number of photons falling per second, per unit area of the pupil.

The total energy per unit for n falling photons is given as:

E = n × 3.96 × 10−19 J s−1 m−2

The energy per unit area per second is the intensity of light.

E = I

n × 3.96 × 10−19 = 10−10


= 2.52 × 108 m2 s−1

The total number of photons entering the pupil per second is given as:

nn × A

= 2.52 × 10× 0.4 × 10−4

= 1.008 × 104 s−1

This number is not as large as the one found in problem (a), but it is large enough for the human eye to never see the individual photons.

Question 11.26:

Ultraviolet light of wavelength 2271 Å from a 100 W mercury source irradiates a photo-cell made of molybdenum metal. If the stopping potential is −1.3 V, estimate the work function of the metal. How would the photo-cell respond to a high intensity (105 W m−2) red light of wavelength 6328 Å produced by a He-Ne laser?


Wavelength of ultraviolet light, λ = 2271 Å = 2271 × 10−10 m

Stopping potential of the metal, V0 = 1.3 V

Planck’s constant, h = 6.6 × 10−34 J

Charge on an electron, e = 1.6 × 10−19 C

Work function of the metal = https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7595/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_242e0bac.gif

Frequency of light = ν

We have the photo-energy relation from the photoelectric effect as:

https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7595/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_242e0bac.gif =  − eV0


Let ν0 be the threshold frequency of the metal.



Wavelength of red light, https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7595/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_686de2b9.gif = 6328 × 10−10 m

Frequency of red light, https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7595/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_m3fb95188.gif


Since ν0νr, the photocell will not respond to the red light produced by the laser.


Question 11.27:

Monochromatic radiation of wavelength 640.2 nm (1nm = 10−9 m) from a neon lamp irradiates photosensitive material made of caesium on tungsten. The stopping voltage is measured to be 0.54 V. The source is replaced by an iron source and its 427.2 nm line irradiates the same photo-cell. Predict the new stopping voltage.


Wavelength of the monochromatic radiation, λ = 640.2 nm

= 640.2 × 10−9 m

Stopping potential of the neon lamp, V0 = 0.54 V

Charge on an electron, e = 1.6 × 10−19 C

Planck’s constant, h = 6.6 × 10−34 Js

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7599/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_242e0bac.gif be the work function and ν be the frequency of emitted light.

We have the photo-energy relation from the photoelectric effect as:

eV0 =  − https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7599/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_242e0bac.gif


Wavelength of the radiation emitted from an iron source, λ' = 427.2 nm

427.2 × 10−9 m

Let https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7599/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_53125279.gifbe the new stopping potential. Hence, photo-energy is given as:


Hence, the new stopping potential is 1.50 eV.


Question 11.28:

A mercury lamp is a convenient source for studying frequency dependence of photoelectric emission, since it gives a number of spectral lines ranging from the UV to the red end of the visible spectrum. In our experiment with rubidium photo-cell, the following lines from a mercury source were used:

λ1 = 3650 Å, λ2= 4047 Å, λ3= 4358 Å, λ4= 5461 Å, λ5= 6907 Å,

The stopping voltages, respectively, were measured to be:

V01 = 1.28 V, V02 = 0.95 V, V03 = 0.74 V, V04 = 0.16 V, V05 = 0 V

Determine the value of Planck’s constant h, the threshold frequency and work function for the material.

[Note: You will notice that to get from the data, you will need to know (which you can take to be 1.6 × 10−19 C). Experiments of this kind on Na, Li, K, etc. were performed by Millikan, who, using his own value of (from the oil-drop experiment) confirmed Einstein’s photoelectric equation and at the same time gave an independent estimate of the value of h.]


Einstein’s photoelectric equation is given as:

eV0 = − https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7603/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_242e0bac.gif



V0 = Stopping potential

h = Planck’s constant

e = Charge on an electron

ν = Frequency of radiation

https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7603/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_242e0bac.gif = Work function of a material

It can be concluded from equation (1) that potential V0 is directly proportional to frequency ν.

Frequency is also given by the relation:


This relation can be used to obtain the frequencies of the various lines of the given wavelengths.



The given quantities can be listed in tabular form as:

Frequency × 1014 Hz






Stopping potential V0






The following figure shows a graph between νand V0.


It can be observed that the obtained curve is a straight line. It intersects the ν-axis at 5 × 1014 Hz, which is the threshold frequency (ν0) of the material. Point D corresponds to a frequency less than the threshold frequency. Hence, there is no photoelectric emission for the λ5 line, and therefore, no stopping voltage is required to stop the current.

Slope of the straight line = https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7603/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_m43fafe3f.gif

From equation (1), the slope https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7603/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_41bf7f4e.gif can be written as:


The work function of the metal is given as:


= 6.573 × 10−34 × 5 × 1014

= 3.286 × 10−19 J

https://img-nm.mnimgs.com/img/study_content/curr/1/12/16/255/7603/NS_5-11-08_Sravana_12_Physics_11_37_NRJ_LVN_html_77890b15.gif= 2.054 eV


Question 11.29:

The work function for the following metals is given:

Na: 2.75 eV; K: 2.30 eV; Mo: 4.17 eV; Ni: 5.15 eV. Which of these metals will not give photoelectric emission for a radiation of wavelength 3300 Å from a He-Cd laser placed 1 m away from the photocell? What happens if the laser is brought nearer and placed 50 cm away?


Mo and Ni will not show photoelectric emission in both cases

Wavelength for a radiation, λ = 3300 Å = 3300 × 10−10 m

Speed of light, c = 3 × 108 m/s

Planck’s constant, h = 6.6 × 10−34 Js

The energy of incident radiation is given as: